Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z104)

The rewrite relation of the following TRS is considered.

c(c(c(a(x1))))	→	d(d(x1))	(1)
d(b(x1))	→	c(c(x1))	(2)
c(x1)	→	a(a(a(a(x1))))	(3)
d(x1)	→	b(b(b(b(x1))))	(4)
b(d(x1))	→	c(c(x1))	(5)
a(c(c(c(x1))))	→	d(d(x1))	(6)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

d^#(b(x1))	→	c^#(x1)	(7)
b^#(d(x1))	→	c^#(x1)	(8)
d^#(x1)	→	b^#(b(b(b(x1))))	(9)
c^#(x1)	→	a^#(a(x1))	(10)
c^#(c(c(a(x1))))	→	d^#(d(x1))	(11)
c^#(c(c(a(x1))))	→	d^#(x1)	(12)
d^#(x1)	→	b^#(b(b(x1)))	(13)
c^#(x1)	→	a^#(a(a(a(x1))))	(14)
c^#(x1)	→	a^#(x1)	(15)
b^#(d(x1))	→	c^#(c(x1))	(16)
d^#(x1)	→	b^#(x1)	(17)
a^#(c(c(c(x1))))	→	d^#(d(x1))	(18)
d^#(b(x1))	→	c^#(c(x1))	(19)
c^#(x1)	→	a^#(a(a(x1)))	(20)
d^#(x1)	→	b^#(b(x1))	(21)
a^#(c(c(c(x1))))	→	d^#(x1)	(22)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(c(c(c(x1))))	→	d^#(x1)	(22)
d^#(x1)	→	b^#(b(b(x1)))	(13)
d^#(x1)	→	b^#(b(x1))	(21)
c^#(c(c(a(x1))))	→	d^#(x1)	(12)
c^#(x1)	→	a^#(a(a(x1)))	(20)
d^#(b(x1))	→	c^#(c(x1))	(19)
a^#(c(c(c(x1))))	→	d^#(d(x1))	(18)
d^#(x1)	→	b^#(x1)	(17)
c^#(c(c(a(x1))))	→	d^#(d(x1))	(11)
c^#(x1)	→	a^#(a(x1))	(10)
d^#(x1)	→	b^#(b(b(b(x1))))	(9)
b^#(d(x1))	→	c^#(x1)	(8)
b^#(d(x1))	→	c^#(c(x1))	(16)
c^#(x1)	→	a^#(x1)	(15)
c^#(x1)	→	a^#(a(a(a(x1))))	(14)
d^#(b(x1))	→	c^#(x1)	(7)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 13
[d(x₁)]	=	x₁ + 85
[b(x₁)]	=	x₁ + 21
[c(x₁)]	=	x₁ + 53
[d^#(x₁)]	=	x₁ + 73
[c^#(x₁)]	=	x₁ + 40
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 9

together with the usable rules

d(x1)	→	b(b(b(b(x1))))	(4)
c(c(c(a(x1))))	→	d(d(x1))	(1)
c(x1)	→	a(a(a(a(x1))))	(3)
b(d(x1))	→	c(c(x1))	(5)
a(c(c(c(x1))))	→	d(d(x1))	(6)
d(b(x1))	→	c(c(x1))	(2)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

a^#(c(c(c(x1))))	→	d^#(x1)	(22)
d^#(x1)	→	b^#(b(b(x1)))	(13)
d^#(x1)	→	b^#(b(x1))	(21)
c^#(c(c(a(x1))))	→	d^#(x1)	(12)
c^#(x1)	→	a^#(a(a(x1)))	(20)
d^#(b(x1))	→	c^#(c(x1))	(19)
a^#(c(c(c(x1))))	→	d^#(d(x1))	(18)
d^#(x1)	→	b^#(x1)	(17)
c^#(c(c(a(x1))))	→	d^#(d(x1))	(11)
c^#(x1)	→	a^#(a(x1))	(10)
d^#(x1)	→	b^#(b(b(b(x1))))	(9)
b^#(d(x1))	→	c^#(x1)	(8)
b^#(d(x1))	→	c^#(c(x1))	(16)
c^#(x1)	→	a^#(x1)	(15)
c^#(x1)	→	a^#(a(a(a(x1))))	(14)
d^#(b(x1))	→	c^#(x1)	(7)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.